最近做大爷们出的训练题，基本每天一道FFT...但是似乎我对FFT的理解只停留在“可以求两个数组的卷积”+背代码阶段，感觉再不重新学一遍要跟不上时代的脚步了...
[12.7]差不多就到这儿吧。。感觉把教材pyx的博客上的每一块都看了一遍，但是感觉做题还是不大会用。。如果以后闲着无聊想自虐的话可能会去写一下策爷的[tex]e^{A_{(z)}}[/tex]。。（不过我感觉以我的智商估计是写不出来了）
1.首先是最基本的FFT：
重新看了一遍算法导论，感觉也不是特别难理解，果然当初学的时候还是太naive了
【BZOJ2194】FFT快速傅立叶
【BZOJ2179】快速傅立叶之二
void fft(atom *x,int n,int fl=1){
for (int i=(n>>1),j=1;j<n;j++){
if (i<j) swap(x[i],x[j]);
int k;
for (k=(n>>1);i&k;i^=k,k>>=1); i^=k;
}
for (int m=2;m<=n;m=(m<<1)){
atom w=(atom){cos(pi*2*fl/m),sin(pi*2*fl/m)};
for (int i=0;i<n;i+=m){
atom cur=(atom){1,0};
for (int j=i;j<i+(m>>1);j++){
atom u=x[j],v=x[j+(m>>1)]*cur;
x[j]=u+v; x[j+(m>>1)]=u-v; cur=cur*w;
}
}
}
}
2.FNT
想起来感觉还是挺显然的，在模意义下用模意义下的单位根替换。
【某模拟题T1】循环计数 mx爷的题，求斯特林数。把斯特林数的那个式子用分治FNT搞就好了（我偷懒空间是[tex]O(n \log n)[/tex]的，稍微改一改就可以做到[tex]O(n)[/tex]
void fft(int *x,int n,int fl=1){
for (int i=(n>>1),j=1;j<n;j++){
if (i<j) swap(x[i],x[j]);
int k;
for (k=(n>>1);k&i;i^=k,k>>=1); i^=k;
}
int now=0;
for (int m=2;m<=n;m<<=1){
int w; now++; if (fl>0) w=G[now]; else w=nG[now];
for (int i=0;i<n;i+=m){
int cur=1;
for (int j=i;j<i+(m>>1);j++){
int u=x[j],v=1ll*x[j+(m>>1)]*cur%mo;
x[j]=(u+v)%mo; x[j+(m>>1)]=(u-v+mo)%mo;
cur=1ll*cur*w%mo;
}
}
}
}
atom solve(int l,int r){
if (l==r){
C[++tot]=l; C[++tot]=1; return (atom){tot-1,tot};
}
int mid=(l+r)>>1; atom k1=solve(l,mid),k2=solve(mid+1,r);
int n=max(mid-l+1,r-mid),len=1;
while ((1<<len)<=(n<<1)) len++;
for (int i=0;i<(1<<len);i++){A[i]=0; B[i]=0;}
for (int i=k1.l;i<=k1.r;i++)A[i-k1.l]=C[i];
for (int i=k2.l;i<=k2.r;i++)B[i-k2.l]=C[i];
fft(A,1<<len); fft(B,1<<len);
for (int i=0;i<(1<<len);i++) A[i]=1ll*A[i]*B[i]%mo;
fft(A,1<<len,-1);
for (int i=0;i<(1<<len);i++) A[i]=1ll*A[i]*two[len]%mo;
atom ans; ans.l=tot+1;
for (int i=0;i<=r-l+1;i++) C[++tot]=A[i]; ans.r=tot;
return ans;
}
3.分治FFT
利用CDQ分治的思想，FFT可以求解形如[tex]f_n=\sum_{i=1}^{n-1}f_i \times A_{n-i}[/tex]的递推式。
【BZOJ3456】城市规划 可以得到递推式[tex]f_n=2^{\binom{n}{2}}- \sum_{i=1}^{n-1}2^{\binom{n-i}{2}} \times \binom{n-1}{i-1} \times f_i[/tex]，意义就是用所有图减去不连通的图的数量，强制1在一个联通块中枚举这个块的大小。之后转化成卷积的形式就可以分治FFT了。
void fft(int *x,int n,int fl=1){
for (int i=(n>>1),j=1;j<n;j++){
if (i<j) swap(x[i],x[j]);
int k;
for (k=(n>>1);k&i;i^=k,k>>=1); i^=k;
}
int now=0;
for (int m=2;m<=n;m<<=1){
int w; now++; if (fl==1) w=G[now]; else w=nG[now];
for (int i=0;i<n;i+=m){
int cur=1;
for (int j=i;j<i+(m>>1);j++){
int u=x[j],v=1ll*x[j+(m>>1)]*cur%mo;
x[j]=(u+v)%mo; x[j+(m>>1)]=(u-v+mo)%mo;
cur=1ll*cur*w%mo;
}
}
}
}
void solve(int l,int r){
if (l==r){
f[l]=(T[l]-1ll*I[l-1]*f[l]%mo+mo)%mo; return;
}
int mid=l+r>>1; solve(l,mid);
int n=max(mid-l+1,r-mid),len=1;
while ((1<<len)<=(n<<1)) len++;
for (int i=0;i<(1<<len);i++){A[i]=0; B[i]=0;}
for (int i=l;i<=mid;i++)A[i-l]=1ll*nI[i-1]*f[i]%mo;
for (int i=1;i<=r-l;i++)B[i]=1ll*T[i]*nI[i]%mo;
fft(A,1<<len); fft(B,1<<len);
for (int i=0;i<(1<<len);i++) A[i]=1ll*A[i]*B[i]%mo;
fft(A,1<<len,-1);
for (int i=mid+1;i<=r;i++) f[i]=(f[i]+1ll*A[i-l]*two[len])%mo;
solve(mid+1,r);
}
4.多项式求逆
看了pyx的博客（感觉pyx的博客已经可以成为中国多项式从入门到提高到进阶三合一通用教材了）后感觉其实如果想到了倍增的话剩下的还是不怎么难想的。。不过似乎这个模型比较奇怪？感觉转化起来有点费劲。似乎满足[tex]f_n=C_{n}-\sum_{i=1}^{n-1}f_i \times A_{n-i} \times B_{n}[/tex]，如果[tex]B_n[/tex]可以把[tex]f_n[/tex]形式转化为[tex]A_{n-i}[/tex]相同的话就可以用这个了。
【BZOJ3456】城市规划
可以得到[tex] \sum_{i=1}^n 2^{\binom{n-i}{2}} \times (n-i)!^{-1} \times (i-1)!^{-1} \times f_i = 2^{\binom{n}{2}} \times (n-1)!^{-1}[/tex]，然后左边可以看成卷积，就可以把一个多项式除过去（就是乘上逆元）。
一直听别人说这玩意常数巨大，跑得还没有分治FFT快。。实际写了一下速度还是快了那么一点点的，但是复杂度的优势完全没了。
void fft(int *x,int n,int fl=1){
for (int i=(n>>1),j=1;j<n;j++){
if (i<j) swap(x[i],x[j]);
int k;
for (k=(n>>1);i&k;i^=k,k>>=1); i^=k;
}
int now=0;
for (int m=2;m<=n;m<<=1){
int w; now++; if (fl==1) w=G[now]; else w=nG[now];
for (int i=0;i<n;i+=m){
int cur=1;
for (int j=i;j<i+(m>>1);j++){
int u=x[j],v=1ll*x[j+(m>>1)]*cur%mo;
x[j]=(u+v)%mo; x[j+(m>>1)]=(u-v+mo)%mo;
cur=1ll*cur*w%mo;
}
}
}
}
void getrev(int n){
if (n==1){
A[0]=quick(C[0],mo-2); return;
}
getrev((n+1)/2); int len=1;
while ((1<<len)<=((n+1)<<1)) len++;
for (int i=0;i<n;i++) B[i]=C[i]; for (int i=n;i<(1<<len);i++) B[i]=0;
fft(A,1<<len); fft(B,1<<len);
for (int i=0;i<(1<<len);i++) B[i]=(2-1ll*A[i]*B[i]%mo+mo)%mo;
for (int i=0;i<(1<<len);i++) A[i]=1ll*A[i]*B[i]%mo;
fft(A,1<<len,-1);
for (int i=0;i<n;i++) A[i]=1ll*A[i]*two[len]%mo;
for (int i=n;i<(1<<len);i++) A[i]=0;
}
5.多项式除法
依然是pyx的博客...关键就是把多项式reverse，后面的就挺好推了。
练习嘛写了一个多项式多点求值愉♂悦一下，写出来之后妈呀怎么10W10W要跑10S..实在是被慢哭了。（策爷：这种常数巨大的东西没前途了
6.多项式开根
仍旧是pyx的博客。。依然是倍增的思想，考虑平方后把式子转化为[tex]A_{(x)}^2 \equiv B_{(x)}(\text{mod}\ x^{n})[/tex]的形式就好了，还是挺好推的。
【codeforces 250E】小朋友和二叉树
接一个方程可以得到母函数的式子是[tex]F_{(x)}=\frac{2}{1+ \sqrt{1-4A_{(x)}}}[/tex]，其中[tex]A_{(x)}[/tex]是那个集合中的数组成的函数。然后就用多项式开根加求逆算出来[tex]F_{(x)}[/tex]就好了。
void fft(int *x,int n,int fl=1){
for (int i=(n>>1),j=1;j<n;j++){
if (i<j) swap(x[i],x[j]);
int k;
for (k=(n>>1);i&k;i^=k,k>>=1); i^=k;
}
int now=0;
for (int m=2;m<=n;m<<=1){
int w; now++; if (fl==1) w=G[now]; else w=nG[now];
for (int i=0;i<n;i+=m){
int cur=1;
for (int j=i;j<i+(m>>1);j++){
int u=x[j],v=1ll*x[j+(m>>1)]*cur%mo;
x[j]=(u+v)%mo; x[j+(m>>1)]=(u-v+mo)%mo;
cur=1ll*cur*w%mo;
}
}
}
}
void getrev(int n){
if (n==1){
rev[0]=quick(fir[0],mo-2); return;
}
getrev((n+1)>>1); int len=1;
while ((1<<len)<=(n<<1)) len++;
for (int i=((n+1)>>1);i<(1<<len);i++) rev[i]=0;
for (int i=0;i<n;i++) A[i]=fir[i];
for (int i=n;i<(1<<len);i++) A[i]=0;
fft(rev,1<<len); fft(A,1<<len);
for (int i=0;i<(1<<len);i++) A[i]=(2-1ll*A[i]*rev[i]%mo+mo)%mo;
for (int i=0;i<(1<<len);i++) rev[i]=1ll*rev[i]*A[i]%mo;
fft(rev,1<<len,-1);
for (int i=0;i<n;i++) rev[i]=1ll*rev[i]*two[len]%mo;
for (int i=n;i<(1<<len);i++) rev[i]=0;
}
void getroot(int n){
if (n==1){
root[0]=1; return;
}
getroot((n+1)>>1); int len=1,m=((n+1)>>1),n2=two[1];
while ((1<<len)<=(n<<1)) len++;
for (int i=0;i<(1<<len);i++){rev[i]=0; fir[i]=0;}
for (int i=0;i<m;i++) fir[i]=root[i];
getrev(n);
for (int i=0;i<(1<<len);i++) A[i]=0;
for (int i=0;i<n;i++) A[i]=C[i];
fft(fir,1<<len); fft(A,1<<len); fft(rev,1<<len);
for (int i=0;i<(1<<len);i++) A[i]=(1ll*fir[i]*n2%mo+1ll*A[i]*rev[i]%mo*n2%mo)%mo;
fft(A,1<<len,-1);
for (int i=0;i<n;i++) root[i]=1ll*A[i]*two[len]%mo;
for (int i=n;i<(1<<len);i++) root[i]=0;
}
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LeanBiome is a cutting-edge approach focused on optimizing the gut microbiome to support weight management and overall health. It is based on the understanding that the trillions of microbes residing in our digestive system play a crucial role in metabolism, nutrient absorption, and immune function. The goal of LeanBiome is to promote a balanced and diverse gut microbiota composition, which is believed to be associated with better weight control and reduced risk of metabolic disorders.https://leanbeome.us
2023年6月29日 22:07
The primary goal of Cortexi is to enhance decision-making processes, automate tasks, and provide valuable insights by leveraging its immense computational power. It has the ability to learn from previous experiences and adapt to new situations, continually improving its performance over time.
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GlucoBerry is a dietary supplement that claims to help regulate blood sugar levels. It is made with a blend of natural ingredients, including maqui berries, Gymnema leaves, chromium, and biotin.
2023年7月10日 22:02
[url=https://glucoberry--glucoberry.com/]GlucoBerry[/url] is a dietary supplement that claims to help regulate blood sugar levels. It is made with a blend of natural ingredients, including maqui berries, Gymnema leaves, chromium, and biotin.
2023年7月10日 22:03
GlucoBerry is a natural supplement that may help to regulate blood sugar levels. However, it is important to talk to your doctor before taking any supplements, especially if you have any underlying health conditions.
2023年7月10日 22:04
GlucoBerry is a natural dietary supplement that helps support healthy blood sugar levels. It is made with a blend of ingredients that have been shown to be effective in lowering blood sugar, including maqui berries, Gymnema leaves, chromium, and biotin. GlucoBerry is safe and effective for most people, and it can be taken alongside other medications or treatments for diabetes.
2023年7月10日 22:04
GlucoBerry is a dietary supplement that claims to help regulate blood sugar levels. It is made with a blend of natural ingredients, including maqui berries, Gymnema leaves, chromium, and biotin. GlucoBerry is said to help support insulin production, reduce sugar cravings, and remove blood sugar spikes.
2023年7月10日 22:04
GlucoBerry is a natural supplement that helps support healthy blood sugar levels.
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2023年7月17日 00:38
With Cortexi supplement you may look forward to a healthy lifestyle that is free of difficulties. Because of its effectiveness in reducing the problems and risks connected with tinnitus and other related disorders, the liquid has gained widespread popularity worldwide. The formula ensures that the core cause of the disease is addressed and that users do not experience age-related reductions in their hearing health.
2023年7月19日 18:09
Discover the wonders of Biossential's 100% certified and organic skincare line. Elevate your self-care routine with nature's finest ingredients. Embrace healthier, more radiant skin today!
2023年7月19日 20:44
Liv pure is a unique medicine that attempts to help the liver lose fat and improve its overall health. It is a natural dietary supplement composed of several natural substances. This supplement is taken on a regular basis in the form of capsules.
2023年7月19日 20:44
Liv Pure is intended to assist you in losing weight and improving your liver function. It works by addressing the root cause of weight gain, which is frequently impaired liver function. The liver is in charge of eliminating toxins from your body, but when it malfunctions, it can cause weight gain and other health issues.
2023年7月20日 18:31
Cortexi is a dietary supplement made with a unique blend of natural substances renowned for their potential advantages in supporting auditory health, cognitive function, and general well-being.
2023年7月20日 18:41
ProDentim oral probiotic supplement is a unique mix of 3.5 billion nutrients and probiotic bacteria that have been shown to work in clinical tests. The Prodentim formula has a simple daily dose and only uses natural ingredients to get the desired result.
2023年7月20日 18:47
ProDentim is a very famous product for tooth health and oral hygiene. It is designed with an amalgam consisting of 3.5 billion probiotics as well as unique nutrients that aid in the maintenance and improvement of the microbiome of your mouth. This amalgam also helps improve your oral health.
2023年7月20日 18:54
Fast Lean Pro's all-natural metabolism booster allows you to shed pounds rapidly without resorting to extreme measures like starvation. Because of the unique combination of components that contain vitamins, minerals, and extracts of herbal plants, your appetite is suppressed, your metabolism is revved up, and you are in the best possible position to shed excess pounds.
2023年7月20日 18:58
Red Boost is an all-natural energy booster for men. It includes potent ingredients that assist to boost your energy and performance, allowing you to enjoy yourself for extended periods of time without experiencing any side effects or pain.
2023年7月22日 21:55
ProvaSlim is a weight loss supplement. It boosts metabolism, enhances gut health, and improves sleep quality.It's like a magic powder that you can take every day. When you use it, you might notice that you look and feel younger and thinner.
2023年7月25日 19:30
Fast Lean Pro is a dietary supplement that aids in weight loss.fastleanproofficial.com/ It is made of natural ingredients and offers an organic solution to stubborn fat that is difficult to lose. It is made in the US in an FDA-approved and GMP-certified lab facility. The process of manufacturing is done by ensuring hygienic and precise conditions while following all the standards. Fast Lean Pro formula comes in powder form and can be easily added to your diet. Fast Lean Pro fat-burning powder does not contain any chemicals or stimulants that might cause side effects such as jitters or stomach irritation.https://fast-leanpro.Pro/
2023年7月25日 19:30
Fast Lean Pro is a dietary supplement that aids in weight loss.fastleanproofficial.com/ It is made of natural ingredients and offers an organic solution to stubborn fat that is difficult to lose. It is made in the US in an FDA-approved and GMP-certified lab facility. The process of manufacturing is done by ensuring hygienic and precise conditions while following all the standards. Fast Lean Pro formula comes in powder form and can be easily added to your diet. Fast Lean Pro fat-burning powder does not contain any chemicals or stimulants that might cause side effects such as jitters or stomach irritation.https://fast-leanpro.Pro/
2023年7月25日 19:31
Endo Peak coulEndo Peak has made its mark in the supplement industry as a ground-breaking item created to boost male health and improve performance.Endo Peak attempts to treat common problems like low libido and stamina issues with its exclusive combination of natural components.d be the answer you’ve been seeking if you’ve been looking for a secure and reliable way to boost your health.
2023年7月25日 19:32
The ingredients in Cortexi work by supporting the health of the inner ear and the auditory nerve. They also help to reduce inflammation and improve blood flow to the ears. If you are looking for a natural way to improve your hearing and cognitive function, Cortexi is a good option to consider.
2023年7月25日 19:33
The creators of Liv Pure pills carefully selected natural ingredients for their proprietary blends, which combine to effectively give a comprehensive weight loss solution.Using Liv Pure, people can successfully achieve their weight loss goals while improving their general health and well-being, claims the company’s official website.
2023年7月25日 19:34
Fast Lean Pro is unlike anything you’ve ever tried or experienced in your life before. It’s the only formula that contains 11 powerful natural ingredients that work in perfect synergy to trick your brain into thinking you are fasting and maintain a healthy weight no matter what and when you eat.
2023年7月25日 19:34
Ikaria Juice is a potent weight reduction dietary supplement that contains a number of potent, scientifically-proven components that can aid in weight loss. It helps people lose extra weight and abdominal fat. Customers of this product can get amazing health advantages.
2023年7月25日 19:35
Cortexi is a 100% safe and secure dietary supplement that promotes ear health by lowering inflammation and mending toxic damage. As a consequence, hearing improves and the ears are protected from additional harm.Cortexi offers more than just hearing aids. Its regular usage promotes attention and focus, sharpens the intellect, and aids in the development of a strong memory. It has a liquid formula, which makes it even easier to use. It is administered orally or sublingually rather than in the ear.
2023年7月25日 19:36
Quietum Plus is a natural supplement that helps combat tinnitus, promotes healthy ears, and enables clearer hearing. The supplement utilizes only the purest natural ingredients, making it a safe and effective option for those seeking to improve their ear health.
2023年7月25日 19:36
Liv Pure was created as a dietary supplement to support the liver. According to the official website, the Liv Pure contains a Purification and Fat-Burning Complex designed to rejuvenate the liver, enhance energy levels, ignites the body's fat-burning furnace, improve metabolic function, and help the user achieve healthy weight loss results. In other words, the Liv Pure formula is built in two steps: first, it revitalizes the liver and stimulates metabolic health, and then, because of this property, weight loss comes as a byproduct of having a clean liver.
2023年7月25日 19:37
EndoPump, the ultimate solution for elevated sensual experiences. Experience enhanced performance, heightened pleasure, and peak sexual health with this premium-quality supplement. EndoPump's natural ingredients work harmoniously to promote normal blood flow, amplify energy levels, and ignite your libido.
2023年7月25日 19:38
Ikaria Juice is a 100% natural weight loss supplement formulated to help you shed pounds safely and effectively. Manufactured under strict FDA standards in the United States, this product adheres to high-quality manufacturing practices to ensure its safety and efficacy.
2023年7月28日 21:04
Cortexi is a groundbreaking, natural hearing support formula making waves in the health industry. Developed by Jonathan Miller, this unique supplement stands apart due to its stimulant-free formulation that focuses diligently on improving ear health and combating hearing problems.
2023年7月29日 19:35
Liv Pure is a premium dietary supplement known for its dual-action of supporting liver health and facilitating weight loss. Developed using scientifically-backed nutrients, it is aimed at revitalizing liver functionality, which in turn, bolsters energy production and fat burning processes.
2023年7月29日 19:35
Quietum Plus can lead to significant improvements in hearing. It aids in the restoration of the delicate, microscopic hairs in the inner ear that play a vital role in translating sound waves into nerve signals.
2023年7月29日 19:36
EndoPump, the ultimate solution for elevated sensual experiences. Experience enhanced performance, heightened pleasure, and peak sexual health with this premium-quality supplement.
2023年7月29日 19:36
Prostadine is specially designed to promote and maintain regular urination, giving you the assurance you need to safeguard your most important organ. Unlike many other supplements, Prostadine does not produce any negative side effects, making it a safe and effective choice for men of all ages.
2023年7月29日 19:36
ProvaSlim Supplement is a special product that helps people with their weight loss goals. It's like a magic powder that you can take every day. When you use it, you might notice that you look and feel younger and thinner.
2023年7月31日 05:30
Health is a vital aspect of well-being, encompassing physical, mental, and emotional balance. Regular exercise, nutritious diet, and sufficient rest are key to maintaining a robust physique. Mental health is equally crucial, necessitating stress management and emotional support. Seeking professional medical advice, routine check-ups, and vaccinations ensure early detection and prevention of illnesses. Prioritizing health empowers individuals to lead fulfilling lives.
2023年7月31日 05:31
Health is a vital aspect of well-being, encompassing physical, mental, and emotional balance. Regular exercise, nutritious diet, and sufficient rest are key to maintaining a robust physique. Mental health is equally crucial, necessitating stress management and emotional support. Seeking professional medical advice, routine check-ups, and vaccinations ensure early detection and prevention of illnesses. Prioritizing health empowers individuals to lead fulfilling lives.
2023年7月31日 05:31
Health is a vital aspect of well-being, encompassing physical, mental, and emotional balance. Regular exercise, nutritious diet, and sufficient rest are key to maintaining a robust physique. Mental health is equally crucial, necessitating stress management and emotional support. Seeking professional medical advice, routine check-ups, and vaccinations ensure early detection and prevention of illnesses. Prioritizing health empowers individuals to lead fulfilling lives.
2023年7月31日 05:49
Quietum Plus is a recently launched supplement designed specifically to improve ear health and prevent age-related hearing loss.
2023年7月31日 05:54
Metabo Flex Is A Revolutionary Plant-Based Supplement Designed To Improve Metabolic Flexibility And Promote Fat Burning.
2023年7月31日 06:41
Fast Lean Pro is a revolutionary dietary formula designed to support the principles of fasting. It is a natural, safe, and effective solution that has already helped numerous individuals achieve remarkable weight loss results. According to its creators, this supplement works by tricking the brain into perceiving a fasting state, prompting the body to burn fat even when not fasting. By mimicking the effects of fasting, Fast Lean Pro assists in maintaining a healthy weight, regardless of eating patterns. It provides a unique approach to long-term weight management and accelerates fat reduction.
2023年7月31日 06:46
EndoPump, a dietary supplement specifically designed to improve the function of your endothelium and support your masculinity. No matter your age, this all-natural product is bound to provide numerous benefits.
2023年7月31日 20:12
Cortexi is a natural supplement designed to support hearing, enhancing auditory function and promoting overall ear health.
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2023年7月31日 20:13
ProDentim is an innovative oral health solution that differentiates itself from other comparable products on the market by providing a special and extremely effective method of preventing dental and oral health issues.
2023年7月31日 20:14
ProDentim! It's not your average dental supplement it's a Advance Formula! Packed with a special mix of 3.5 billion probiotics and essential nutrients, ProDentim has the power to take care of your teeth and gums like never before.
2023年7月31日 21:26
Dentiox Pro is a supplement that helps repair injured gums, strengthen teeth and remove plaque from the surface of the teeth. It can also help control the bad breath by maintaining existing bacteria on their way out or killing off any new ones before they take hold. It has been proven scientifically as the dentist's most prevalent oral health product because it fights harmful substances inside the teeth and outside. There's more room for improvement in areas such as stains caused by coffee. It aids in the removal of inflammation, removes plaque and bacteria from the gum and teeth, contains essential nutrients to rebuild damaged gums and teeth. Dentitox Pro is an innovative dental or oral health formula that promises to improve the health of your teeth and gums. Dentitox Pro is formulated with all-natural components obtained from local cultivators, which have been researched and tested for their effectiveness in maintaining dental health. Dentitox Pro is natural supplement that can ensure healthy gums, fresh breath, and optimal tooth health.Dentitox Pro drops are formulated with a unique blend of organic plant-based ingredients that have been clinically proven to treat periodontal disease. The best part about Dentitox Pro is that it is safe for use by older people or those who have lost dental strength. This is because the Dentitox Pro drops do not contain any harsh chemicals that can damage the teeth and gums.
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Keraessentials is a natural oil that helps prevent toenail fungus, dry skin, smelly feet, and yellow, weak nails. When you use Keraessentials oil every day, you will see great benefits that last a long time. Kerassentials oil also has things that protect from damage and fight swelling. These help soothe and calm painful nails and scars.
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Amiclear, the leading-edge blood sugar supplement backed by top research and a powerful blend of ingredients, harmoniously working together to naturally balance glucose levels and promote healthy weight loss.
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2023年8月09日 22:04
Liv Pure is a dietary supplement specifically crafted to provide a holistic approach towards health and wellness, with a distinct focus on natural weight loss and liver purification. The core philosophy behind the product is to offer a clean, safe, and effective pathway to weight loss without the strenuous demands typically associated with traditional methods like strict dieting and rigorous workouts.
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The PowerBite is the only formula that supports teeth and gums and incorporates a special mineral blend as a convenient dental candy.
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2023年8月11日 22:16
Power Bite is the #1 Rated Oral health Support Formula that mainly targets fresh breath, improving teeth power, and supporting gum health.
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2023年8月11日 22:16
Power Bite combines the effectiveness of mineral supplementation with the convenience of delicious candy. It offers a delightful way to support healthy teeth and gums, making oral care a pleasurable experience for users of all ages.
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2023年8月11日 22:16
PowerBite is the top candy of dental innovation, strictly crafted to elevate your oral care. Crafted proudly in the USA within an FDA-certified facility, PowerBite embodies the essence of purity and safety. What truly sets it apart is its revolutionary approach – a keen emphasis on mouth and body mineralization. This unique formula imparts unparalleled dental strength, guards against erosion, and boosts gum health. Revel in the newfound confidence of fresher breath and fortified teeth, all achieved without a trace of negative side effects. Transform your dental care routine with PowerBite today, and embrace a brighter, healthier smile.
2023年8月11日 22:16
ProDentim your ultimate solution for a healthier smile. Packed with special ingredients, ProDentim is the perfect oral supplement for strong teeth and gums.
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2023年8月11日 22:17
PowerBite offers a treasure house of benefits beyond its dental courage. Indulging in these toothsome candies not only strengthens teeth, but also supports overall mouth health by promoting balanced pH levels. With the focus on mineralization, Power Bite aids in refilling essential minerals for your body. Additionally, its natural ingredients contribute to a healthier oral microbiome, promoting a harmonious environment in your mouth. Elevate your oral care with Power Bite's comprehensive approach, and enjoy the wholesome advantages it brings to your overall well-being.
2023年8月12日 23:19
One business man predicts that if nothing is done and the Health insurance premiums keep increasing that in the year 2008, the amount of health premium contribution to employer will surpass their profit. Professionals within and outside the field of Florida health insurance, think that the reason for increase in Florida health insurance premium rates are due to many factors, such as high administration expenditure, inflation, poor or bad management, increase in the cost of medical care, waste etc. implants
2023年8月13日 12:32
Cortexi is a renowned brand dedicated to crafting natural formulations that promote vibrant cognitive senstivity and sustained auditory vitality throughout your golden journey. Meticulously tailored to address the distinct needs of those aspiring to uphold mental and auditory well-being as time gracefully unfolds, Cortexi's remarkable formula seamlessly weaves together premium elements sourced globally.
2023年8月14日 00:05
The PowerBite is the only formula that supports teeth and gums and incorporates a special mineral blend as a convenient dental candy.
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2023年8月14日 00:06
Power Bite is the #1 Rated Oral health Support Formula that mainly targets fresh breath, improving teeth power, and supporting gum health.
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2023年8月14日 00:06
Power Bite combines the effectiveness of mineral supplementation with the convenience of delicious candy. It offers a delightful way to support healthy teeth and gums, making oral care a pleasurable experience for users of all ages.
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2023年8月14日 00:06
PowerBite is the top candy of dental innovation, strictly crafted to elevate your oral care. Crafted proudly in the USA within an FDA-certified facility, PowerBite embodies the essence of purity and safety. What truly sets it apart is its revolutionary approach – a keen emphasis on mouth and body mineralization. This unique formula imparts unparalleled dental strength, guards against erosion, and boosts gum health. Revel in the newfound confidence of fresher breath and fortified teeth, all achieved without a trace of negative side effects. Transform your dental care routine with PowerBite today, and embrace a brighter, healthier smile.
2023年8月14日 00:07
ProDentim your ultimate solution for a healthier smile. Packed with special ingredients, ProDentim is the perfect oral supplement for strong teeth and gums.
Experience it yourself ProDentim supports gum health, strengthens enamel, and fights harmful bacteria for fresh, lasting breath.
2023年8月14日 00:07
PowerBite offers a treasure house of benefits beyond its dental courage. Indulging in these toothsome candies not only strengthens teeth, but also supports overall mouth health by promoting balanced pH levels. With the focus on mineralization, Power Bite aids in refilling essential minerals for your body. Additionally, its natural ingredients contribute to a healthier oral microbiome, promoting a harmonious environment in your mouth. Elevate your oral care with Power Bite's comprehensive approach, and enjoy the wholesome advantages it brings to your overall well-being.
2023年8月19日 21:30
Exipure was officially launched to the public on October 21, 2021. Since then, there have been discussions, questions, and speculations about this new weight loss supplement.
2023年8月19日 21:31
Cortexi Hearing Support Formula is a health supplement produced from the purest natural ingredients that are intended to improve your hearing as well as your attention and memory by boosting brain health.
2023年8月19日 21:31
Digital marketing is a field that involves promoting products, services, or brands through various digital channels and technologies, such as search engines, social media, email, mobile apps, and websites. The goal of digital marketing is to reach and engage with potential customers online, build brand awareness, generate leads, and ultimately drive sales and revenue.
2023年8月19日 21:32
DuoTrim is an advanced weight loss support formula that contributes to significant weight loss without drastic side effects. The supplement is unique worldwide because it uses a unique combination of chemical ingredients designed to target an abnormally low core body temperature.
2023年8月19日 21:33
DuoTrim is an advanced weight loss support formula that contributes to significant weight loss without drastic side effects. The supplement is unique worldwide because it uses a unique combination of chemical ingredients designed to target an abnormally low core body temperature.
2023年8月19日 21:33
Flexafen is a revolutionary joint health formula designed to provide temporary relief for occasional aches and discomfort. It contains a unique blend of time-tested nutrients, these ingredients aim to support collagen and synovial fluid in the joints, which are crucial for joint health. By doing so, Flexafen helps improve mobility, build strength, lubricate joints, and provide lasting comfort.
2023年8月19日 21:34
Fast Lean Pro is a type of powdered dietary supplement that can help you lose weight in a healthy way. It also supports the renewal of your body's cells and helps with fasting and metabolism. Each scoop of Fast Lean Pro contains special, all-natural ingredients that have been shown in studies to promote healthy weight loss. These ingredients can effectively accelerate the processes of autophagy (cell renewal) in your body. By using the weight-loss supplement Fast Lean Pro, you can burn calories while still maintaining good health.
2023年8月19日 21:35
Fast Lean Pro is a type of powdered dietary supplement that can help you lose weight in a healthy way. It also supports the renewal of your body's cells and helps with fasting and metabolism. Each scoop of Fast Lean Pro contains special, all-natural ingredients that have been shown in studies to promote healthy weight loss. These ingredients can effectively accelerate the processes of autophagy (cell renewal) in your body. By using the weight-loss supplement Fast Lean Pro, you can burn calories while still maintaining good health.
2023年8月19日 21:35
Gut Vita ™ Differs Anything You have actually Ever Before Attempted Before. It Is among The Only Products In The World With A Proprietary Blend Of The Highest-Quality Plant Kingdom And Also Fiber That Research Study Has Shown To Enhance Digestive Health And Wellness And Gut Microbiome For Both Women And Also Guy.
2023年8月19日 21:36
Gut Vita ™ Differs Anything You have actually Ever Before Attempted Before. It Is among The Only Products In The World With A Proprietary Blend Of The Highest-Quality Plant Kingdom And Also Fiber That Research Study Has Shown To Enhance Digestive Health And Wellness And Gut Microbiome For Both Women And Also Guy.
2023年8月19日 21:36
Liv Pure is a revolutionary supplement that targets the root cause of weight gain and stubborn belly fat in individuals of any gender. The formula is designed to improve the functioning of the liver, which plays a crucial role in weight loss. The liver is responsible for filtering toxins and chemicals from the body, but it also serves as the body's primary fat-burning furnace. This means that the liver determines whether the food we eat should be converted into energy or stored as fat.
2023年8月19日 21:37
Liv Pure is a revolutionary supplement that targets the root cause of weight gain and stubborn belly fat in individuals of any gender. The formula is designed to improve the functioning of the liver, which plays a crucial role in weight loss. The liver is responsible for filtering toxins and chemicals from the body, but it also serves as the body's primary fat-burning furnace. This means that the liver determines whether the food we eat should be converted into energy or stored as fat.
2023年8月19日 21:37
Liv Pure is a revolutionary supplement that targets the root cause of weight gain and stubborn belly fat in individuals of any gender. The formula is designed to improve the functioning of the liver, which plays a crucial role in weight loss. The liver is responsible for filtering toxins and chemicals from the body, but it also serves as the body's primary fat-burning furnace. This means that the liver determines whether the food we eat should be converted into energy or stored as fat.
2023年8月19日 21:38
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Neotonics is a special type of probiotic supplement that's gaining a lot of attention. People are excited about it because it's good for the friendly bacteria in our tummies and might even help us look younger. Here's the cool part: it tackles the things that cause our skin to age.
Crafted solely from natural elements, this supplement is a treasure trove of nourishing nutrients. It's like a rich mix of vitamins, minerals, collagen, and probiotics that not only nurture the gut's balance but also create a harmonious blend of benefits for promoting supple and adaptable skin.
2023年11月28日 04:43
https://alphatonic.uk/
Alpha Tonic is a natural dietary supplement designed to boost men's virility and overall well-being. Packed with powerful Himalayan nutrients, this powdered supplement aims to address issues like low energy, reduced muscle mass, and other signs of decreased virility that many men may experience. In this blog, we'll delve into what Alpha Tonic offers and how it claims to work wonders for men's health.
What sets Alpha Tonic apart is its unique blend of eleven natural ingredients. These ingredients include vitamins, minerals, and herbal extracts that have been traditionally associated with promoting healthy virility levels. From Boron that aids in regulating virility levels to Ashwagandha known for its stamina-boosting effects, these components work together to address various aspects of male health.
2023年11月28日 05:11
Serolean operates on a well-defined mechanism to assist individuals in their weight loss journey effectively. At its core, Serolean's formulation focuses on enhancing metabolism, promoting fat burning, and optimizing energy levels. Let's delve into the intricacies of how Serolean works to support your quest for a healthier and fitter body.
By incorporating scientifically proven ingredients like green tea extract, caffeine anhydrous, L-carnitine, CLA (Conjugated Linoleic Acid), and cayenne pepper extract, Serolean tackles weight management from various angles.
2023年11月28日 16:11
Puravive, the latest addition to the ever-expanding world of dietary supplements, claims to offer a myriad of health benefits, from improved immune function to enhanced overall wellness. As consumers, we're continually bombarded with promises of better health, and it's only natural to be skeptical.
In this in-depth and objective review, we aim to unravel the mysteries behind Puravive, shedding light on its ingredients, potential benefits, and any associated risks. By the end, you'll be equipped with the knowledge needed to make an informed decision about whether this supplement deserves a place in your daily routine.
2023年11月28日 16:49
"Puravive: Unveiling an Innovative Approach for Effective Weight Management"
Uncover a holistic approach to achieving your weight loss goals with Puravive, a natural supplement designed to support healthy weight and enhance your overall well-being. Our commitment to quality and science ensures you embark on a journey towards a healthier lifestyle.
2023年11月28日 19:33
SeroLean is a weight management supplement designed to support individuals on their journey to a healthier weight. It accomplishes this through several key mechanisms. First, it helps control appetite, making it easier to resist overeating and manage portion sizes. By curbing cravings, it assists in reducing overall calorie intake.
Additionally, SeroLean may boost metabolism, enhancing the body's ability to burn calories efficiently. This can be especially beneficial when combined with a balanced diet and regular physical activity. Moreover, the supplement can provide an energy boost, combating the fatigue that often accompanies lifestyle changes associated with weight loss.
2023年11月29日 03:53
Supplement is a dietary product designed to support weight management and overall health. It's important to note that this supplement should be considered as part of a broader approach to health and should not be seen as a magical solution for weight loss.
The strength of PuraVive lies in its use of natural ingredients. These ingredients have been chosen for their potential to assist the body in maintaining a healthier weight. Some of the components include Luteolin, Kudzu Root, Holy Basil, Oleuropein, White Korean Ginseng (Panax), Amur Cork Bark, Propolis, and Quercetin. These ingredients work together to target stubborn fat and improve metabolic processes.
2023年11月29日 03:55
Supplement is a dietary product designed to support weight management and overall health. It's important to note that this supplement should be considered as part of a broader approach to health and should not be seen as a magical solution for weight loss.
The strength of PuraVive lies in its use of natural ingredients. These ingredients have been chosen for their potential to assist the body in maintaining a healthier weight. Some of the components include Luteolin, Kudzu Root, Holy Basil, Oleuropein, White Korean Ginseng (Panax), Amur Cork Bark, Propolis, and Quercetin. These ingredients work together to target stubborn fat and improve metabolic processes.
2023年11月29日 04:40
PuraVive Supplement primarily focuses on brown adipose tissue (BAT), often referred to as brown fat. Brown fat is considered healthier and can help in burning calories more efficiently. This supplement aims to enhance the body's ability to store fat as brown fat, promoting a slimmer and more energetic body.
PuraVive is available in easy-to-swallow capsule form. It's essential to follow the recommended dosage, which is typically one capsule with a glass of water before breakfast. The supplement is generally safe to use, with limited reported side effects. However, as with any dietary product, it's wise to consult with a healthcare professional before incorporating it into your routine, especially if you have pre-existing medical conditions or are taking other medications.
2023年11月29日 04:45
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2023年11月29日 04:45
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